Fractional inclusions of superquadraticity via multiplicative calculus

In this paper, we introduced a novel class of superquadraticity, termed multiplicatively (superquadratic interval-valued functions) superquadratic $\mathcal{I}\mathcal{V}\mathcal{F}$s and investigated their unique properties using interval order relations and (Riemann-Liouville) $\mathcal{R}\mathcal{.}\mathcal{L}$-fractional integral operators. By employing the framework of multiplicative calculus, we established new fractional integral inequalities specifically of Hermite-Hadamard ( $\mathcal{H}\mathcal{.}\mathcal{H}$) type, for these superquadratic $\mathcal{I}\mathcal{V}\mathcal{F}$. Furthermore, we extended our analysis to derive fractional inequalities for the product and quotient of multiplicatively superquadratic and subquadratic $\mathcal{I}\mathcal{V}\mathcal{F}$ functions within the same calculus setting. By setting $\ell =1$, the results naturally reduced to their corresponding integer-order forms for multiplicatively superquadratic $\mathcal{I}\mathcal{V}\mathcal{F}$. To validate our theoretical findings, we present numerical computations and graphical illustrations based on several illustrative examples, showcasing the practical utility and robustness of the results. In addition, we explored potential applications of these inequalities, particularly in the context of linear combinations of special means. This provides a fresh perspective on superquadratic $\mathcal{I}\mathcal{V}\mathcal{F}s$ and expands the scope of multiplicative convex analysis. The results presented in this work are entirely new within the framework of fractional multiplicative calculus and have not been previously reported in the literature. We believe that this study will pave the way for future research, offering a deeper understanding of convexity phenomena and powerful tools for mathematical modeling involving $\mathcal{I}\mathcal{V}\mathcal{F}s$.